Find Reduced Row Echelon Form [[1,0,1],[2,1,4],[-1,3,6]]

$[\begin{array}{c} 1 & 0 & 1 \\ 2 & 1 & 4 \\ - 1 & 3 & 6 \end{array}]$

Step 1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 \\ 2 - 2 \cdot 1 & 1 - 2 \cdot 0 & 4 - 2 \cdot 1 \\ - 1 & 3 & 6 \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 2 \\ - 1 & 3 & 6 \end{array}]$

Step 2

Perform the row operation $R_{3} = R_{3} + R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 2 \\ - 1 + 1 \cdot 1 & 3 + 0 & 6 + 1 \cdot 1 \end{array}]$

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 3 & 7 \end{array}]$

Step 3

Perform the row operation $R_{3} = R_{3} - 3 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & 7 - 3 \cdot 2 \end{array}]$

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array}]$

Step 4

Perform the row operation $R_{2} = R_{2} - 2 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 \\ 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 5

Perform the row operation $R_{1} = R_{1} - R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0 & 0 - 0 & 1 - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$